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Author: D’Antona, Ottavio M Munarini, EmanueleBook Title: Advances in applied mathematics.Journal Article: The Cycle-Path Indicator Polynomial of a Digraph.Year: 2000Volume: v 25 n 1 JUL 01Pages: 41Location:Rush: onNeeded By:User Name: OAOE-Mail address: elucas@goliat.espol.edu.ecPhone #: 20-12-00

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Advances in Applied Mathematics 25, 41-56 (2000)doi:l0.1006/aama.2000.0689, available online at http:/W.idealibrary.com on IDE@@

The Cycle-Path Indicator Polynomial of a Digraph

Ottavio M. D’Antona

Dipartimento di Scienze dell’lnformazione, Universik degli Studi di Milano,Via Comelico, 39, 20135 Milano, Italy

and

Emanuele Munarini

Dipartimento di Matematica, Politecnico di Milano, Pza Leonardo da Vinci, 32,20133 Milano, Italy

Received February 21, 2000; accepted March 24, 2000

A cycle-path cover of a digraph D is a spanning subgraph made of disjoint cy-cles and paths. In order to count such covers by types we introduce the cycle-path indicator polynomial of D. We show that this polynomial can be obtained bya deletionxontraction recurrence relation. Then we study some specializations ofthe cycle-path indicator polynomial, such as the geomettic coverpo&nomial (the ge-ometric version of the cover polynomial introduced by Chung and Graham), thecycle cover polynomial, and the path cover polynomial.

As an application, we consider chromatic arrangements of non-attacking rooksand the associated polynomial which is symmetric and generalizes the usual rookpolynomial. 0 2000 Academic Press

Key Words: cycle-path covers; cycle-path indicator polynomial; geometric coverpolynomial; cycle cover polynomial; path cover polynomial; a-polynomial; coverpolynomial; chromatic arrangements of non-attacking rooks; generalized rook poly-nomial.

CONTENTS

1. Introduction.2. The cycle-path indicator polynomial.3. The geometric cover polynomial.4. The recurrence relation for the cycle-path indicator polynomial.5. The cycle cover polynomial and the path cover polynomial.6. Chromatic arrangements of non-attacking rooks.7. Concluding remarks.

41

0AP 0196~8858lOO $35.00Copyright 0 2000 by Academic Press

All rights of reproduction in any form reserved.

42 D’ANTONA AND MUNARINI

1. INTRODUCTION

In [3], Chung and Graham introduced the cover polynomial of a digraphand showed that this polynomial, as well as the Tutte polynomial of a graph,satisfies a deletion-contraction recurrence relation and that several otherpolynomials can be obtained as specializations of this one. In [2], Chowgeneralized the cover polynomial to a symmetric function as Stanley didfor the chromatic polynomial in [12] and proved that the latter polynomialcan be obtained by the former.

In this paper we introduce the cycle-path indicator polynomial of a di-graph D in order to classify the cycle-path covers of D by types. (As usualindicator suggests types.) This polynomial is in many indeterminates (but itis not symmetric) and yields many polynomials related to a digraph such asthe characteristic and permanental polynomial.

As a natural specialization of the cycle-path indicator polynomial wehave the geometric cover polynomial, that is, the geometric version of thecover polynomial of Chung and Graham that allows us to enumerate cycle-path covers according only to the total number of cycles and paths. Thesetwo polynomials are quite similar since both satisfy the same deletion-contraction recurrence and are determined just by different initial condi-tions.

From this remark the question if such recurrence, or something alike,also holds for the cycle-path indicator polynomial immediately follows. Thishope seems doomed to failure in the category of digraphs because of themeaning of the indices of the indeterminates. However, by enlarging thecategory of digraphs to that of weighted digraphs, we are able to find adeletion-contraction recurrence relation also for the cycle-path indicatorpolynomial.

We also introduce the cycle cover and the path cover polynomials of D,both specializations of the cycle-path cover polynomial.

Finally, as an application, we consider chromatic arrangements of non-attacking rooks on a board B, where the principal sub-boards determinedby the rooks of the same color are disjoint. The associated polynomial issymmetric and generalizes the usual rook polynomial [ll].

2. THE CYCLE-PATH INDICATOR POLYNOMIAL

In this paper a digraph D is always a finite directed multi-graph on nvertices. As usual V(D) and E(D) are respectively the set of vertices andthe set of edges of D. We denote by 0 the digraph with no edges and novertices and by D, + D2 the sum (i.e., disjoint union) of the two digraphsD1 and D,.

CYCLE-PATH INDICATOR 43

In the sequel all cycles and paths will be directed (even without mention).We write $B[D] for the set of directed cycles and paths of the digraph D.Notice that there are no cycles on 0 vertices and that any single vertex is apath of length 0.

A cycle-path cover of a digraph D is a spanning subgraph of D formed bya (vertex-) disjoint union of directed cycles and paths. A block of a cycle-path cover is any of its cycles or paths. Equivalently, a cycle-path cover ofD is a pair (7r, $) where v is a partition of the set of vertices of D and+= (+B)BW is a family of structures where $B is a spannig cycle or pathof the subgraph of D induced by the block B of T. We denote by %[D] theset of all cycle-path covers of D.

Let us introduce some notations. We set [0] := 0 and [n] := { 1,2 n} for any n 2 1.A’l;il2h2...nhn and p,

We write x for the n-tuple (x,, x2, . . . , x~). If= lk@2 . . . nkn are two partitions of n, h t- n and

,z l- n, then we set x,, := x:1$2 . . . xi’ and h @ ,u := lhl+k12h2+k2 . . . nhn+km.We define the weight function Ind: .B[D] + [x, y], where N[x, y] is the

semiring of polynomials in the indeterminates x1, x2, . . . and yl, y2, . . . withcoefficients in N, by associating xk to any cycle y on k vertices (of lengthk) and yk to any path 7r on k vertices (of length k - 1).

A cycle-path cover is of type (h, p) if it has hi cycles on i vertices and kjpaths on j vertices. Clearly, if (h, p) is the type of a cycle-path cover of D,then A @ p t- n.

Now we can extend the weight Ind to the set %[D] by setting

Ind(C) := n Ind(P).PEC

(1)

Therefore, if (h, ,u) is the type of C, we have

Ind( C) = xAyp = xfl e . . xi”$ . . - yn”“.

The indicator polynomial of a digraph D is the weight of the set %[D]with respect to the weight function Ind; that is

I(D;x;y) : = C Ind(C) = c i+(D) xAyp, (2)C&[D] A@/.+n

where ihp(D) is the number of cycle-path covers of D of type (h, p).The indicator polynomial satisfies the following obvious properties.

PROPOSITION 1. For any digraphs D, D1, and D2:

1. I(0; x; y) = 1;

2. I(D1 + D,; x; y) = I@,; x; y)I(D2; x; y);3. if D1 is isomorphic to D2, then I(D,; x; y) = I(D2; x; y);

44 D’ANTONA AND MUNARINI

4. if D* is the digraph obtained by reversing the direction of the edges ofD, then I(D*; x; y) = I(D; x; y).

The following is a useful property we shall often use in the sequel.

PROPOSITION 2. The cycle-path indicator polynomial enjoys the followinghomogeneity property:

I(D; pxl, p2+, . . . , P”x,; PYI, p2y2, - - - 9 P”Y,)= p”‘(D; ~1, ~2, . . . , xn; ~1, ~27 . . - 7 v,)- (3)

ProoJ Every monomial in the indicator polynomial is of the form

hl 4, kl knXhyCL = X1 * * * xn yl * . * yn 7

where l(h, + k,) + 2(h2 + k2) + . e a + n(h, + k,) = n. Therefore, replacingxi with p’xi and yi with /JYj ( 1 I i, j 5 n), we obtain the monomial pnxAyCL.I

Let A4 be a monoid. We say that a weight function w: .%[D] + A4 isuniform on cycles and paths if it depends only on their number of vertices.In this case zu is equivalent to the two n-tuples c = (c,, . . . , cn) and p =(Pl, * - * 3 p,), where ck = w(y) for a cycle y on k vertices and pk = w(n)for a path v on k vertices. The function w can be extended to %[D] bysetting, for every C E %[D],

w(C) := n w(P).PCC

Let P,(D) be the weight associated to the set %[D] with respect to thefunction w. Therefore

C,,(D) := c w(C) = c n w(P);CE8[D] CE%[D] pee

that is, by the uniformity of zu,

Ku(D) = c iA, JD)C*P,.ACBph

Hence we have the following

LEMMA 1. If w is a uniform weight on 9[ D] defined by c = ( c1 , . . . , c,)andp=(pI,... , p,), then the weight P,(D) of the set %[D] is given by

P,(D) = I(D; c; p). (4)

The two next propositions give examples in which related digraphs haverelated cycle-path indicator polynomials.

CYCLE-PATH INDICATOR 45

PROPOSITION 3. If Do is the digraph obtained from D by adding a loop toevery vertex, then

I(D”; x1, x2, . . . , xn; ~1, ~2, - . - 3 Yn)

= I(D; x1, x2, . . . , x,; x1 + ~1, ~‘2, - - - 7 Yn). (5)

Pro05 A cycle-path cover of D” is equivalent to a cycle-path cover ofD endowed with a subset of the set of the paths on one vertex. This subsetdeterminates which loops in the cover of D” come from the new ones. u

PROPOSITION 4. Let D be a digraph on n vertices, and let DIk] be thedigraph obtained from D by substituting each edge with k parallel edges. Then

I (DIk];

ProoJ The cycles and the paths of Dfk] are on the same vertices as thoseof D, and for any cycle (path) of length h of D there are kh cycles (paths)of Dtk] with the same length on the same vertices. Therefore

I(D’kl; XI, ~2, . . . , x,; yl, ~2, . . . , yn)

= I(D; kxl, k2x2, . . . , knx,; yl, ky2, . . . , k”-‘y,).

So from the homogeneity property (3), (6) follows. m

PROPOSITION 5. If A is the adjacency matti of a digraph D, then we havethe following identities :

1. I(D; -1, . . . , -1; 0, . . . , 0) = (-1)” det A,

2. I(D; -1, . . . , -1; p, 0, . . . , 0) = det(p1 - A),

3. I(D;l,.,., l;O ,..., O)=perA,

4. I(D; 1, . . . , 1; p, 0, . . . , 0) = per (PI + A),

5. I(D; -1, 1, . . . , (-1)“; p, 0,. . . , 0) = per (PI - A).

Proof We begin by proving the second identity. We recall [5] that if thecharacteristic polynomial of the digraph D is

det(pl - A) = e an-kpk,k=O

then we have

aj = 1 (-l)y(c) ,CErj[D]

46 D’ANTONA AND MUNARINI

where T,[D] is the set of the subdigraphs of D with exactly j vertices andformed by vertex-disjoint cycles, and y(C) is the number of cycles in C.Therefore we have

I(D; -1, . . . , -l;p, 0, . . . , 0) = C iA,(D)(-l)hl . . a (-l)hnpklOkZ . . *Ok,h@pFn

= 5 C ihlk (D)( -l)hl+“‘+hnpk

k=O AFn-k

= q c

k = O Ccr,,-,[D]

(-1)Y’“‘)pk

= 2 an-kpk

k=O

= det(pl - A).

Similarly, if

Per(Pl + A) = k bnmkpk,k=O

then bj = IIj[D]]. Thus, proceeding as before, we obtain the fourth identity.Finally, using (3) we have

per (pl - A) = (-l)“per (-pl + A)

= (-l)“I(D; 1, . . . ) 1; -p, 0, . . . , 0)

= I(D; -1, 1, . . . , (-l)n; p, 0,. . . ) 0). I

EXAMPLES. (1) Let 1, be the digraph with n vertices and no edges.Then

I(In; x; Y) = YT- (7)

(2) Let N, be the digraph with n vertices and a loop on each vertex.Since N, = I;, by Proposition (3) and identity (7), we have

I(Nn;x;Y) = (~1 + Yl)“. (8)(3) Let K; be the complete digraph of order n, that is, the digraph

such that V(K;) := [n] and E(K;) := [n] x [n]. Its indicator polynomial isgiven by

I(Ki;x;y) = c 25he&n ‘A$, xAyp’

(9)

CYCLE-PATH INDICATOR 47

where zA := lh12h2 . . a nhnh1!h2! m. e h,! and z’, := k,!k,! . . . k,!. The expo-nential formal series of such polynomials is given by

c wq; XT Y) 2n?O

. =exP(z(2 +Yn)un)*

-(10)

Indeed, a cycle-path cover of Kz is equivalent to a partition 7~ of I/(Ki) inwhich each block is endowed with a spanning cycle or a spanning path ofthe subgraph of Ki induced by the block itself. Since the subgraph of Kiinduced by any k-subset of vertices is isomorphic to K& then the weight(with respect to the function Ind) of the set formed by the spanning cyclesand the spanning path of such a graph is given by the polynomial Ik :=(k - l)!xk + k!yk. Therefore

I(K;; x; y) =?I

k1

’ ’ ’ ’ ’l,..., n ,..., n

lk,+...+nk,=n

where there are ki i’s as lower indices in the multinomial coefficient. Ex-panding every 1Fi, it is straightforward to obtain (9). It is also straightfor-ward to see that the exponential formal series of I(Ki; x, y) is exactly (10).Finally, notice that if yj = 0 for all j, then (10) reduces to the well knownexponential formal series for the numbers of permutations of given type [4].

3. THE GEOMETRIC COVER POLYNOMIAL

We now introduce a polynomial which classifies the cycle-path covers ofa digraph D by the total number of cycles and paths, disregarding theirtype. The geometric cover polynomial of a digraph D is defined by

C(D; X, y) := C Ci,j(D)X’y’, (11)i, jr0

where ci,j(D) is the number of cycle-path covers of D with i cycles and jp a t h s .

PROPOSITION 6. The geometric cover polynomial of a digraph D is given by

C(D;x,y) = I(D;x, . . . . x;y,. . . ,y). (12)

The geometric cover polynomial of a digraph is very similar to the coverpolynomial, introduced by Chung and Graham in [3],

E(D; X, y) = C ci, j(D)X’y’,i, j?O

48 D’ANTONA AND MUNARINI

where yi := y(y - 1). . . (y - j + 1) are the falling factorial polynomials (in[3] notations are slightly different).

These two polynomials satisfy the same deletion-contraction recurrencerelation. To see this we need some more definitions. For any vertex v of D,D \ v is the digraph obtained from D by removing the vertex v and everyedge incident with it. For any edge e of D, D \ e is the digraph obtainedfrom D by removing the edge e. Finally, if e is an edge of D going fromv1 to v2, then D/e is the digraph obtained from D by removing the edge e,all the edges entering ul, and all the edges leaving u2, and by identifying qand v2 in a single vertex u.

For the geometric cover polynomial we have the following recurrence:

[LOOPS] if e is a loop on the vertex V, then

C(D; X, y) = x C(D \ v; x, y) + C(D \ e; x, Y), (13)

[EDGES] if e is an edge but not a loop, then

C(Q x, Y) = C(D \ e; x, Y) + V/e; x, y), (14)with the initial condition:

[I.c.] if D has n vertices and no edges, then

C(D; x, y) = yn. (15)

The cover polynomial of Chung and Graham [3] satisfies the same recur-rence relation but the initial condition is F(D; x, y) = ya for any digraphD with n vertices and no edges.

4. THE RECURRENCE RELATION FOR THE CYCLE-PATHINDICATOR POLYNOMIAL

The geometric cover polynomial and the cover polynomial satisfy thesame recurrence because they are associated with the same combinatorialobjects, i.e., the cycle-path covers of a digraph. Therefore it is reasonableto ask if there exists a similar recurrence even for the cycle-path indicatorpolynomial. However, it seems hard to find such a recurrence in the cate-gory of digraphs, since in the contraction of an edge we lose the essentialinformation of the lengths of cycles and paths containing that edge.

To recover this information we have to change category. In contractingan edge e we remove e (and possibly other edges) and identify its endpointsin a single vertex. The central idea is to distinguish those vertices comingfrom a contraction by giving them a weight. Two vertices will be identifiedin a single vertex with a bigger weight where the total weight will be the

CYCLE-PATH INDICATOR 49

number of edges contracted in it. In this way we are led to the concept ofweighted digraphs. We now formalize this idea.

A weighted digraph is a digraph D endowed with a function (or weight)w: V(D) + N defined on the vertices of D. We write D, for the weighteddigraph (D, w) .

We now extend the definition of the cycle-path indicator polynomial to aweighted digraph D,. Cycles, paths, and cycle-path covers of D, are thoseof D. We write SB[D,] for the set of cycles and paths of D,, and write%[D,] for the set of cycle-path covers of D,. The function Ind: .B[D,] +N[x, y] is defined in the following way: if p is a cycle or a path on k verticesvl,...,vk, then

if p is a cycle

if p is a path,

where p := w(q) + e . . + W(Q) + k. We extend the function Ind to %[D,,,]by setting

Ind(C) := n Ind(P).PC

The cycle-path indicator polynomial of D, is the weight of %[D,] withrespect to the function Ind; that is

Ind(D,;x;y) := c Ind(C) = c ihP(Dw) xAyP. (16)CWD”,l A7l.L

Let us introduce some other definitions. For any vertex v of D,, D, \ vis the weighted digraph (D \ v, w’) where the weight w’ is the restriction ofw to the new set of vertices. For any edge e of D,, D, \ e is the weighteddigraph (D \ e, w). Finally, if q and v2 are the endpoints of an edge eof D,, then D,/e is the weighted digraph (D/e, w*) where the weight w*is defined by

w*(v) := 1w(v) ifv$u

4Jl) + NV,) + 1 if v = u,

where u is the vertex in which vl and v2 are identified.We can now state and prove our main result.

THEOREM 1. The cycle-path indicator polynomial of a weighted digraphD, is determined by the recurrence relation:

[LOOPS ] if e is a loop on the vertex v, then

w,; xi Y) = ~,(v)+l I(&, \ v; x; Y) + W, \ e; x; Y),

[E D G E S ] if e is an edge but not a loop, then

W,,; x; Y) = I(& \ e; x; Y) + W&,/e; x; Y), (18)

50 D’ANTONA AND MUNARINI

with the initial condition:[I.c.] if D, has no edges, then

ww; x; Y) = n Yw(lJ)+l * (1%VEV(D)

Prooj If e is a loop on the vertex v, then we have the bijection

according to the fact that the loop is either a block of a cycle-path cover ofD, or not.

On the other hand, if e is an edge but not a loop, then we have thebijection

according to the fact that either e is an edge of a block in a cycle-path coverof D, or not.

Finally, if D, has no edges, then it has the unique cover in which eachblock is a path of length zero.

Hence, considering cardinalities, we obtain the stated recurrence. I

In order to obtain the cycle-path indicator polynomial of a digraph wecan think of it as the weighted digraph with zero weight everywhere andthen apply the above recurrence.

There follows an example, where the digraphs stand for their cycle-pathindicator polynomials:

= y1(. l +q) + q. q +*2) + (⌧3 + 02)

= YlCYf + Y2) + 2(YlY2 + Y3) + (x3 + Y3)

= x3 + Y: + 3YlY2 + 3Y3.

CYCLE-PATH INDICATOR 51

5. THE CYCLE COVER POLYNOMIAL AND THE PATH COVERPOLYNOMIAL

We now introduce two polynomials which are a specialization of the ge-ometric cover polynomial. A cycle couer of a digraph is a cycle-path coverwithout paths, while a path couer is a cycle-path cover without cycles.

The cycle cover polynomial of a digraph D is the polynomial

S(D; t) := t s,(D)tk, (20)k=O

where Sk(D) is the number of the cycle cover of D with k cycles. Thispolynomial is a special case of the circuit poZynomiaZ introduced in [6]. (See[7, 81 for applications to determinants and permanents.)

The path cover polynomial of a digraph D is the polynomial

LCD; t) := 2 &(D)tk, (21)k=O

where Zk(D) is the number of the path covers of D with k paths. Thispolynomial is a special case of the partition polynomial studied by Wagnerin [13] and has itself already been studied there for simple digraphs.

We call Sk(D) and lk(D) the Stirling numbers and the Lah numbers of thedigraph D, respectively. These names are justified by the fact that when Dis the complete digraph Ki then Sk(Ki) and lk(Ki) are the usual (signless)Stirling numbers of the first kind and the usual (signless) Lah numbers,respectively. In particular S(Ki; t) = t”, where t” := t(t + 1). . a (t + n - 1)are the rising factorial polynomials, and

Notice that L(Ki; -t) are the Laguerre polynomials [lo].

PROPOSITION 7. The cycle cover and path cover polynomials of a digraphD can be obtained by the cover polynomial and by the indicator polynomialby the identities:

S(D;t)=C(D;t,O)=I(D;t ,..., t;O ,..., 0), (22)

L(D;t) = C(D;O, t) =I(D;O ,..., 0;t ,..., t). (23)

The cycle cover polynomial of a digraph is the product of the cycle coverpolynomials of its strongly connected components, and the path cover poly-nomial of a digraph is the path cover polynomial of the digraph with itsloops removed. Moreover, if D is acyclic, then C(D; t, t) = L(D; t).

We recall that an articulation point is a vertex whose removal increasesthe number of connected components.

52 D’ANTONA AND MUNARINI

PROPOSITION 8. Let D, +V D, be the digraph in which the subgraphs D,and D, are articulated in v. Then

S(D1 + D2; t

>= S(D1 \ v; t)S(D,; t) + S(D1; t)S(D, \ v; t) . (24)v

Proofi In any cycle cover of D1 +V D2, the vertex v belongs either to acycle of D1 or to a cycle of D2. In the first case we have the cycle coversof D, + (D2 \ v), while in the second case we have the cycle covers of(4\74+D2- 1

PROPOSITION 9. Let D be a digraph made of two digraphs D, and D2joined only by the switch (T = { ( v1 , v2), ( v2, q)}, where v1 is a vertex of D1and v2 is a vertex of D2; then

S(D; t) = S(D,; t)S(D,; t) + tS( D, \ q; t)S(D, \ v2; t) . (25)

ProoJ The cycle covers of D containing the switch CT are in a bijectivecorrespondence with the cycle covers of D, \ v1 + D, \ v2. On the otherhand, the cycle covers of D not containing the switch u are in a bijectivecorrespondence with the cycle covers of D, + D,. 1

Let (P, 5) be a poset, and let Inc(P) be the incomparability graph ofP, where the vertices are the elements of P and two vertices v and w areadjacent when they are incomparable (v -$ u) and w $ v). Let D, be thedigraph associated to the relation 5, that is, the digraph such that T/(D,) =P and E(D,) = {(x, y) E P x P : x 5 y}, and let DL be the digraphobtained from D, by removing every loop. Clearly 0: is a transitive acyclicdigraph without multiple edges. A path in 0: is equivalent to a chain inP and to an independent subset of Inc(P) (that is, a subset in which notwo vertices are adjacent). Therefore a path cover in 0: is equivalent to adecomposition of P into chains and to a partition into independent blocksof Inc(P). It follows that

a(Inc(P); t) = L(Dl; t),

where a(Inc(P); t) is the a-polynomial of Inc(P) [l]. We recall that thea-polynomial of a graph G is defined by

a(@ t) = 2 p,dG)tk,k=O

where pk(G) is the number of partitions of V(G) in k independent blocks.As an example let us consider a chain C,+i = [n] with n elements. Then

Inc(C,-i) is 1, and 0: is the transitive tournament T,, that is, the digraphsuch that V(T,) := [n]-and E(T,) := {(i, j) E [n] x [n] : i -C j}. So we have

cr(1,; t) = L(T,; t) = Sri(t) := 2 ; xk,k=O 0

CYCLE-PATH INDICATOR 53

where the ,!&(t) are the exponential polynomials [lo]. Thus the Lah num-bers Z,(T,J of a transitive tournament are the usual Stirling numbers of thesecond kind 1;) [9].

As a further example, let us consider the poset P, = {x0, x1, . . . , xn}where x0 5 xi for i = 1, . . . , n are the unique nontrivial relations. ThenInc(P,) = K, + K, and 0: is the tree in which x0 is a source. Therefore-

a(Inc(P,); t) = L(D>; t) = tn+l + ntn = P(t + n).-

Now let Q, be the sum of n - 1 copies of P,. Then

cT(Inc(Q,); t) = tn2-“-1A,(t; -l),

where the A,( t; -1) = t(t + n)n-l are Abel polynomials [lo].

6. CHROMATIC ARRANGEMENTS OF NON-ATTACKING ROOKS

An n-board B is a subset of [n] x [n] and can be represented by an n x nmatrix MB = [b,,i] where bi,j = 1 if (i, j) E B and b!,j = 0 if (i, j) 6 B. LetD, be the digraph having MB as its adjacency matrix.

An arrangement of k non-attacking rooks on B is a subset of B, whoseelements are thought of as chess rooks, in which there are no two rookson the same line (row or column). Such an arrangement corresponds toa cycle-path cover of D, with k edges. More precisely, a rook on the cell(i, j) of the board B corresponds to the edge from the vertex i to the vertexj of Dg.

A principal sub-board of order k of B is the subset of B obtained by theintersection of k rows and k columns with the same index. For instance, ifwe consider the board

then the set {C&2), C&4), (4,2), (4,4)} is a principal sub-board but forthe set { (2, l), (2,2), (3, l), (3,2)} it is not the case.

Now suppose having rooks of h different colors. A chromatic arrangementof k non-attacking rooks on B is an arrangement of k non-attacking coloredrooks for which there exist disjoint principal sub-boards of B such that every

54 D’ANTONA AND MUNARINI

principal sub-board contains only rooks of the same color. Of the followingtwo examples (where the R’s and the S’s are rooks of different color)

H m

only the first is a chromatic arrangement of non-attacking rooks (eventhough the second is a non-attacking arrangement of rooks disregardingcolors).

The generazized rook polynomial of a board B is the polynomial

R(B; t) := &(B)t,, (26)h

where A = 1’12~2 . . . hih is a partition and r,(B) is the number of chromaticarrangements of non-attacking rooks on the board B with ij rooks of thejth color (1 5 j < h). For h = 1 we have the usual [ll] rook polynomialof B:

R(B; t) := c Q(B)&k?O

Since chromatic arrangements of non-attacking rooks depend only onthe number of colors involved, but not on their labels, the generalized rookpolynomial (26) is symmetric.

THEOREM 2. The generalized rook polynomial of a board B can be ob-tained by the cycle-path indicator polynomial of the digraph D, by the identity

W; t) = I(&; PI, ~2,. . . , pn; PO, PI,. . . , pn-I), (27)

where pk(tI, . . . , th) := t; + t; + * * . + tf if k 2 1 and po( tI, . . . , th) := 1are the power sum symmetric functions.

Prooj A chromatic arrangement of rooks on B is equivalent to a cycle-path cover C of D, in which to each block of C with at least one edge isassigned one of the available colors. A cycle on k vertices, that is, of lengthk, gives a contribution of k rooks, while a path on k vertices, that is, oflength k - 1, gives a contribution of k - 1 rooks.

Therefore the generalized rook polynomial R(B; t,, t2, . . . , th) is theweight of %[DB] with respect to the function w defined by

w(p)= t +*-e+tIk hk if p is a cycle on k verticestk-1 + . . . + p

1 if p is a path on k vertices

if k 2 1 and by w(p) = 1 if p is a path on 0 vertices. Since w is auniform weight on B[DB] defined by the vectors c = (pI, . . . , p,) andP = (PO, * * * , pnml), then from Lemma 1 (27) follows. 1

CYCLE-PATH INDICATOR 55

From Theorem 2 we can write the generalized rook polynomial in termsof the power sum symmetric functions pk,

w; t) = c C,P, ,A

(28)

where p,, = p; . - . pi and the coefficient c,, is the number of the cycle-pathcover of DB with ij blocks of length j, for j = 1,2, . . . , n.

When h = 1 we have

R(B; t) = I(D,; t, t2, . . . , P; 1, t, . . . , P-l),

that is, using the homogeneity property (3) of the cycle-path indicator poly-nomial and the definition of the geometric cover polynomial,

R(B; t) = PC D,; 1, f .( >

(29

7. CONCLUDING REMARKS

In Proposition 1 we have seen that the digraphs D and D* have the samecycle-path indicator polynomial. In general these digraphs are not isomor-phic, as it happens for the two digraphs - a n d tiVm . so,non-isomorphic digraphs can have the same cycle-path indicator polyno-mial. However, D and D* are anti-isomorphic. Hence two natural questionsarise: are there any non-isomorphic and non-anti-isomorphic digraphs withthe same cycle-path indicator polynomial? How can the digraphs having thesame cycle-path indicator polynomial be characterized?

REFERENCES

1. E Brenti, G. E Royle, and D. G. Wagner, Location of zeros of chromatic and relatedpolynomials of graphs, Canad. .I. Math. 46 (1994), 55-80.

2. T Y. Chow, The path-cycle symmetric function of a digraph, Adu in Math. 118 (1996),71-98.

3. E R. K. Chung and R. L. Graham, On the cover polynomial of a digraph, .I. Combin.Theory Ser: B 65 (1995), 273-290.

4. L. Comtet, “Advanced Combinatorics,” Reidel, Boston, 1974.5. D. M. Cvetkovic, M. Doob, and H. Sachs, “Spectra of Graphs. Theory and Applications,”

Academic Press, New York, 1980.6. E. J. Farrell, On a class of polynomials obtained from the circuits in a graph and its

applications to characteristic polynomials of graphs, Discrete Math. 25 (1979), 121-133.7. E. J. Farrell, J. W. Kennedy, and L. V. Quintas, Graph permanent, determinant, and cycle

polynomials, Bull. ICA 23 (1998), 73-82.8. E. J. Farrell, J. W. Kennedy, and L. I? Quintas, Permanents of matrices: A graph cycle

polynomial approach, Bull. ICA 28 (2000), 99-106.

56 D’ANTONA AND MUNARINI

9. R. L. Graham, D. E. Knuth, and 0. Patashnik, “Concrete Mathematics,” Addison-Wesley,Reading, MA, 1989.

10. R. Mullin and G.-C. Rota, On the foundation of combinatorial theory III. Theory ofbinomial enumeration, in “Graph Theory and Its Applications,” (B. Harris, Ed.), pp. 167-213, Academic Press, New York, 1970. Reprinted in “Gian-Carlo Rota on Combinatorics”(J. P. S. Kung, Ed.), Birkauser, Boston, 1995.

11. J. Riordan, ‘An Introduction to Combinatorial Analysis,” Wiley, New York, 1958.12. R. Stanley, A symmetric function generalization of the chromatic polynomial of a graph,

Adu Math. 111 (1995), 166194.13. D. Wagner, The partition polynomial of a finite set system, J. Combin. Theory Sel: A 56

(1991), 138-159.